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Electricity and Magnetism
Drift Velocity and Current Density in Conductors
Drift velocity is the average velocity attained by charged particles, such as electrons in a material, due to an electric field. It forms the microscopic foundation for macroscopic electrical conduction in metals.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of drift velocity, relation between current and drift velocity, Ohm's law verification via microscopic parameters, and factors affecting drift velocity.
PGT extension
Derivation of drift velocity from relaxation time, microscopic form of Ohm's law (J = sigma E), mobility of charge carriers, and temperature dependence of resistivity.
Prerequisites
Electrostatics in conductors
Thermal motion of electrons in metals
Basic vector calculus and kinematics
Microscopic Origin of Current
In the absence of an electric field, free electrons in a conductor move randomly with thermal velocities of the order of 105 m/s, resulting in zero net current.
When an external electric field is applied, electrons experience an electrostatic force and acquire a slow net directional motion opposite to the field.
The magnitude of this drift velocity is extremely small, typically of the order of 10-4 m/s under normal laboratory current conditions.
Relaxation Time and Mean Free Path
During their drift, electrons undergo continuous collisions with the fixed positive ion cores of the metal lattice.
The average time interval between two successive collisions is called the relaxation time, denoted by the symbol τ.
The mean free path (λ) is the average distance traveled by an electron between successive collisions.
Mobility of Charge Carriers
Mobility (μ) is defined as the magnitude of drift velocity per unit electric field.
It serves as a measure of how quickly a charge carrier can move through a metal or semiconductor under the influence of an electric field.
The SI unit of mobility is m2/V·s.
Microscopic Form of Ohm's Law
Current density (J) is directly proportional to the applied electric field (E) inside a uniform conductor.
The constant of proportionality is the electrical conductivity (σ), expressed as $J = \sigmaE$.
This relation represents the local or microscopic form of Ohm's law, independent of the geometry of the conductor.
Derivations
Derivation of Drift Velocity and Current Relation
Conditions: Uniform conductor of cross-sectional area A, length l, and electron number density n.
Consider an electric field E applied across the conductor, causing an acceleration $a = eE/m$ for each electron of mass m and charge e.
The drift velocity vd is the average velocity gained between collisions, given by vd = a$\tau = (eE/m)\tau$.
The total number of free electrons in a segment of length vdΔt is $N = n A v_{d} \Deltat$.
The charge passing through the cross-section in time Δt is Δ$Q = Ne = n A e v_{d} \Deltat$.
The electric current $I = \DeltaQ/\Deltat = n e A v_{d}$.
Result: $I = n e A v_{d}$
Derivation of Conductivity and Microscopic Ohm's Law
Conditions: Free electron gas model in a metal under steady electric field E.
Substitute the expression for drift velocity vd = (eE/m)τ into the current equation $I = n e A v_{d}$.
We get $I = n e A (eE/m)\tau = (n e^{2} \tau / m) A E$.
Current density $J = I/A = (n e^{2} \tau / m) E$.
Compare this with the microscopic Ohm's law $J = \sigmaE$ to identify the conductivity σ.
Result: $\sigma = n e^{2} \tau / m$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Drift Velocity Expression
$$v_d = \frac{e E \tau}{m}$$
e = electronic charge, E = electric field, \tau = relaxation time, m = mass of electron m/s
Valid for steady electric fields in uniform conductors
Current in terms of Drift Velocity
$$I = n e A v_d$$
n = free electron density, e = charge, $A = cross-sectional area$, $v_d = drift velocity$ A
Uniform cross-section and steady current
Current Density
$$J = \frac{I}{A} = n e v_d$$
J = current density, I = current, A = area A/m2
Uniform current distribution across area A
Mobility
$$\mu = \frac{v_d}{E} = \frac{e \tau}{m}$$
\mu = mobility, $v_d = drift velocity$, E = electric field m2/V\cdot s
Linear response regime
Microscopic Ohm's Law
$$J = \sigma E$$
J = current density, \sigma = conductivity, E = electric field A/m2 (J), S/m (\sigma), V/m (E)
Isotropic and homogeneous medium
Electrical Conductivity
$$\sigma = \frac{n e^2 \tau}{m}$$
n = electron density, \tau = relaxation time S/m (Siemens per meter)
\alpha = temperature coefficient of resistivity, \rh$o_0 = initial resistivity$ Ω·m
Valid over moderate temperature ranges for metals
Graphs and experiments
Drift Velocity vs Electric Field
Draw: A straight line passing through the origin with a positive slope equal to mobility μ on a vd versus E plot.
Interpret: Shows that drift velocity is directly proportional to the applied electric field in standard metallic conductors.
Exam use: Frequently tested in conceptual multiple-choice questions regarding the linearity of carrier transport.
Resistivity vs Temperature for Metals
Draw: A curve that starts near zero absolute temperature at a residual resistivity value and increases nearly linearly at higher temperatures.
Interpret: As temperature increases, thermal agitation of lattice ions decreases relaxation time τ, thereby increasing resistivity ρ.
Exam use: Asked in PGT questions relating microscopic parameters to macroscopic temperature coefficients.
Current Density vs Electric Field (Microscopic Ohm's Law)
Draw: Linear plot of J versus E where the slope represents electrical conductivity σ.
Interpret: Confirms obedience to microscopic Ohm's law for ohmic conductors.
Exam use: Evaluates understanding of field-current density relationships.
Solved numericals
TGT core
A copper wire of cross-sectional area 2 × 10-6 m2 carries a current of 3.2 A. If the free electron density of copper is 8 × 1028 m-3, calculate the drift velocity of electrons.
Identify given values: I = 3.2 A, $A = 2 \times 10^{-6} m^{2}$, $n = 8 \times 10^{28} m^{-3}$, $e = 1.6 \times 10^{-19} C$.
The relaxation time of free electrons in a metal is 2 × 10-14 s. Find the mobility of electrons in the metal given that the mass of an electron is 9.1 × 10-31 kg and its charge is 1.6 × 10-19 C.
A potential difference of 4 V is applied across a 2 m long metallic wire. If the mobility of electrons is 4.5 × 10-3 m2/V·s, find the drift speed of the electrons.
Calculate electric field $E = V / l = 4 V / 2 m = 2 V/m$.
The Drude model assumes free electrons behave like an ideal gas and undergo instantaneous collisions with stationary ion cores, which neglects electron-electron interactions.
Common misconceptions
Common error: Drift velocity is the speed at which electric signals or current travel through a wire.
Correct: Electric signals travel at a significant fraction of the speed of light, whereas drift velocity of electrons is extremely slow (~10-4 m/s).
The electric field propagates nearly instantaneously through the conductor, setting all free electrons in motion simultaneously, which establishes current immediately.
Common error: Switching on a light switch causes an electron from the switch to reach the bulb instantly.
Correct: Individual electrons drift very slowly; the energy is transmitted via electromagnetic field waves propagating through the circuit.
Drift speed is microscopic, but the coordinated movement of the sea of electrons driven by the electric field constitutes the electric current.
Quick revision
Drift velocity v_d is proportional to the applied electric field E and relaxation time τ.
Drift Velocity Expression: $v_d = \frac{e E \tau}{m}$
Current in terms of Drift Velocity: $I = n e A v_d$
Must remember
Random thermal velocities of free electrons in a conductor are of the order of $10^5\text{ m/s}$, whereas their drift velocity is extremely small, typically of the order of $10^{-4}\text{ m/s}$.
The microscopic form of Ohm's law relates current density $J$ and electric field $E$ as $J = \sigma E$, where $\sigma$ is electrical conductivity.
Mobility $\mu$ is defined as the magnitude of drift velocity per unit electric field, given by $\mu = \frac{v_d}{E} = \frac{e \tau}{m}$.
Electrical resistivity is inversely proportional to relaxation time and free electron density: $\rho = \frac{m}{n e^2 \tau}$.
Electric signals propagate through a conductor at a speed close to the speed of light, while the actual drift speed of electrons is very slow.
Glossary
Drift Velocity ($v_d$)The average uniform speed with which free electrons get drifted towards the positive terminal under the influence of an applied electric field.
Relaxation Time ($\tau$)The average time interval between two successive collisions of free electrons with the positive ion cores in a metal lattice.
Mobility ($\mu$)The magnitude of drift velocity acquired per unit electric field, serving as a measure of how quickly a charge carrier moves.
Current Density ($J$)The electric current flowing per unit cross-sectional area normal to the current, expressed as $J = n e v_d$.
Mean Free Path ($\lambda$)The average distance traveled by a free electron between two successive collisions.
Thermal Motion vs Drift Motion
Parameter
Thermal Motion
Drift Motion
Cause
Thermal energy at room temperature
Applied external electric field
Velocity Magnitude
Order of $10^5\text{ m/s}$
Order of $10^{-4}\text{ m/s}$
Net Current
Zero net current
Non-zero net current
Macroscopic vs Microscopic Ohm's Law
Feature
Macroscopic Ohm's Law
Microscopic Ohm's Law
Expression
$V = IR$
$J = \sigma E$
Dependency
Depends on conductor geometry (length, area)
Independent of geometry, intrinsic property
Variables involved
Voltage, Current, Resistance
Current Density, Conductivity, Electric Field
Practice Quiz
Choose an option to reveal the explanation.
1. What is the typical order of magnitude of the thermal velocity of free electrons in a metal at room temperature?
Explanation: In the absence of an electric field, free electrons move randomly with thermal velocities of the order of $10^5\text{ m/s}$ due to thermal agitation.
2. What is the order of magnitude of drift velocity of electrons in a metal wire under normal laboratory current conditions?
Explanation: The drift velocity of electrons is extremely small under typical conditions, generally of the order of $10^{-4}\text{ m/s}$.
3. Which relation correctly expresses current $I$ in terms of free electron density $n$, cross-sectional area $A$, charge $e$, and drift velocity $v_d$?
Explanation: The macroscopic current is related to microscopic parameters by the equation $I = n e A v_d$, derived by considering the volume of electrons passing a cross-section per unit time.
4. The microscopic form of Ohm's law is given by which of the following expressions?
Explanation: The microscopic form of Ohm's law relates current density $J$ directly to electric field $E$ via conductivity $\sigma$ as $J = \sigma E$.
5. How is mobility $\mu$ of charge carriers defined in terms of drift velocity $v_d$ and electric field $E$?
Explanation: Mobility is defined as the magnitude of drift velocity per unit electric field, $\mu = \frac{v_d}{E}$.
6. What is the SI unit of mobility?
Explanation: Since mobility is $\mu = \frac{v_d}{E}$, its unit is $\frac{\text{m/s}}{\text{V/m}} = \text{m}^2\text{/V}\cdot\text{s}$.
7. Which of the following represents the expression for electrical conductivity based on the Drude model?
Explanation: From the Drude model derivation, electrical conductivity is given by $\sigma = \frac{n e^2 \tau}{m}$.
8. If the relaxation time $\tau$ of free electrons in a conductor decreases with an increase in temperature, how does electrical resistivity change?
Explanation: Resistivity is given by $\rho = \frac{m}{n e^2 \tau}$. As temperature increases, thermal agitation increases, decreasing relaxation time $\tau$, which causes resistivity $\rho$ to increase.
9. What does the relaxation time $\tau$ physically signify?
Explanation: Relaxation time is defined as the average time interval between two successive collisions of free electrons with lattice ion cores.
10. Why do electric signals travel through a circuit almost instantaneously while electron drift velocity is very slow?
Explanation: The electric field propagates through the conductor at a speed close to the speed of light, causing all free electrons across the circuit to drift simultaneously, establishing current immediately.
11. Current density $J$ can be expressed in terms of drift velocity $v_d$ as:
Explanation: Current density $J = \frac{I}{A} = \frac{n e A v_d}{A} = n e v_d$.
12. Which of the following statements about drift velocity is correct?
Explanation: Drift velocity is given by $v_d = \frac{e E \tau}{m}$, showing it is inversely proportional to mass $m$ and directly proportional to $E$ and $\tau$, directed opposite to the field.
13. The temperature coefficient of resistivity $\alpha$ for metals is generally:
Explanation: For metals, resistivity increases with temperature, meaning their temperature coefficient of resistivity $\alpha$ is positive.
14. Which parameter is an intrinsic property of a material and independent of its physical dimensions?
Explanation: Conductivity ($\sigma$) and resistivity ($\rho$) are intrinsic material properties, whereas resistance depends on geometry ($l/A$).
15. If the electric field applied across a conductor is doubled, what happens to the drift velocity of the free electrons (assuming constant temperature)?
Explanation: Drift velocity is directly proportional to the electric field ($v_d = \frac{e E \tau}{m}$), so doubling $E$ doubles $v_d$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If $n$ is the number of free electrons per unit volume, $e$ is the electronic charge, $A$ is the cross-sectional area, and $v_d$ is the drift velocity, then the electric current $I$ is given by:
Explanation: The fundamental relation between current and drift velocity is $I = n e A v_d$.
2. The microscopic form of Ohm's law is expressed as:
Explanation: Microscopic Ohm's law relates current density $J$ and electric field $E$ via conductivity $\sigma$ as $J = \sigma E$.
3. The SI unit of electrical mobility is:
Explanation: Mobility $\mu = \frac{v_d}{E}$ has units of $\frac{\text{m/s}}{\text{V/m}} = \text{m}^2 / \text{V}\cdot\text{s}$.
4. Which of the following correctly gives the expression for electrical resistivity in terms of relaxation time $\tau$ and electron density $n$?
Explanation: Conductivity is $\sigma = \frac{n e^2 \tau}{m}$, and resistivity is the reciprocal: $\rho = \frac{1}{\sigma} = \frac{m}{n e^2 \tau}$.
5. What happens to the relaxation time of electrons in metals when temperature increases?
Explanation: As temperature increases, thermal vibrations of lattice ions increase, leading to more frequent collisions and hence a decreased relaxation time $\tau$.
6. The drift velocity of free electrons in a conductor is of the order of:
Explanation: Drift velocity is extremely slow under normal lab currents, typically of the order of $10^{-4}\text{ m/s}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If a current of $4.8\text{ A}$ flows through a wire of cross-sectional area $10^{-4}\text{ m}^2$ and the free electron density is $10^{28}\text{ m}^{-3}$, what is the drift velocity of the electrons? (Take $e = 1.6 \times 10^{-19}\text{ C}$)
Explanation: Using $I = n e A v_d$, we get $v_d = \frac{I}{n e A} = \frac{4.8}{10^{28} \times 1.6 \times 10^{-19} \times 10^{-4}} = \frac{4.8}{1.6 \times 10^5} = 3 \times 10^{-5}\text{ m/s}$.
2. What is the relation between mobility $\mu$, electronic charge $e$, relaxation time $\tau$, and mass $m$ of an electron?
Explanation: Since $v_d = \frac{e E \tau}{m}$ and $\mu = \frac{v_d}{E}$, it follows that $\mu = \frac{e \tau}{m}$.
3. Which of the following parameters does NOT affect the drift velocity of electrons in a uniform conductor carrying a steady current?
Explanation: Drift velocity $v_d = \frac{e E \tau}{m}$ depends on electric field $E$, relaxation time $\tau$, and mass $m$, but is independent of the total length of the conductor for a given uniform field.
4. If the electric field inside a conductor of conductivity $2 \times 10^7\text{ S/m}$ is $1.5\text{ V/m}$, what is the magnitude of the current density?
Explanation: Using microscopic Ohm's law $J = \sigma E = (2 \times 10^7\text{ S/m}) \times (1.5\text{ V/m}) = 3.0 \times 10^7\text{ A/m}^2$.
5. How does the electrical resistivity of semiconductors change with an increase in temperature?
Explanation: In semiconductors, an increase in temperature increases the number density of charge carriers $n$ significantly, outweighing the decrease in relaxation time, which causes resistivity to decrease.
6. If the radius of a uniform wire carrying a steady current is halved while keeping the current constant, what happens to the drift velocity of the free electrons?
Explanation: Since $I = n e A v_d$ and $A = \pi r^2$, if radius $r$ is halved, area $A$ becomes one-fourth. For constant $I$, $v_d$ must be quadrupled.
7. What is the physical meaning of current density $J$?
Explanation: Current density $J$ is defined as current per unit cross-sectional area ($J = I/A$) oriented perpendicular to the direction of current flow.
8. Which of the following correctly describes the dimensions of electrical mobility?
9. In a metallic conductor, if the temperature is raised, the drift velocity of electrons for a constant applied electric field will:
Explanation: Since $v_d = \frac{e E \tau}{m}$ and temperature increase decreases relaxation time $\tau$, the drift velocity will decrease for a constant electric field $E$.
10. Which statement best contrasts thermal velocity and drift velocity?
Explanation: Thermal motion is random and averages to zero net current, whereas drift motion is a slow net directional movement caused by an electric field that produces a macroscopic current.